## HomeWork ONE

1. 1.3

a , where machine epsilon , is the base of the system and is lowest possible value of exponent.

b with being maximum value of exponent,

c

2. 1.8

a

Here we divided numerator and denominator by to avoid overflow.

b

Here we divided numerator and denominator by to avoid overflow.

c

d

Here we divided numerator and denominator by to avoid overflow.

e

Calculate

Since , and , and , and using the fact that even number raised to the power of 10 is even, and odd number raised to teh power of 10 is odd, we obtain

3. 1.9 Homer states

Is it true?

a If Horner right,

should be 0, yet Matlab returns

b If Horner is right,

should be zero, yet Matlab returns .

c If Horner is right,

should be , yet Matlab returns

d If Horner is right,

should be zero, yet Matlab returns

e The product of two digit numbers may have up to digits. Therefore may have up to 48 digits, which is longer then Matlab's 16 digits. Therefore we can not use doubles in Matlab to either prove or disprove this conjecture.

Note that can be explicitly calculated on a computer with inifinite precision (check out vpa command in matlab), because it is an integer. Using the “sym” command, or Mathematica allows us to get the answer:

4. 1.11 Here we deal with

for small values of . Expanding we obtain

This is very close to the example we studied in class, so all information can be taken directly from lecture notes
5. 1.13 Here we study for small the function

Taking into account that

the numerator should be equal to for small . Yet, the numerator is computed by first and then taking the logarithm. Therefore for small values of the numerator is approximately given by

we therefore may use information from lectures to characterize behaviour of this function.

6. 1.17 a

Use Taylorâ€™s theorem to show that

 (31)

Denoting and using

we obtain

Since , we obtain (32).

b Why

 (32)

This is due to “rounding to nearest”, is rounded to , and is either or . In other words, the distance between a number and its floating point representation is less than half a distance between nearest two floaing point numbers.

c

Since and , we get . More precisely, . Use this together with (32) and (33) to get

 (33)

d This question implies that

 (34)

If this property is not satisfied, then the result can not be obtained.

To obtain this result, write

then use (35).

Author's note: in (d) the right hand side shoule be and in (e) the right hand side should be .

e The RHS can be rewritten as

Now use to obtain the desired result.

f Using we get . Picture shows that accuracy is obtained at , i.e. . In other words the estimate is optimistic.